1. Why were there polar bears on the Island?
The zoological research arm of the DHARMA Initiative brought many creatures to the Island in order to study the development of animal life in this exotic climate. What they found was that, miraculously, even animals from vastly different habitats (such as Ursus maritimus) flourished.

2. Why did women have trouble conceiving on the Island?
All life stems from the Island's life-force. When it was damaged in the 1977 Incident, the ability for the Island to give life was damaged as well.

3. What do the Egyptian markings represent?
Countless civilizations visited the Island over the millennia. Just like the DHARMA Initiative and the 815 survivors, the Egyptians were just another group of people who discovered the Island, tried exploit/understand it, and ended up becoming its protectors as a result.

4. What's the deal with the cork?
Just like the Swan was to the DHARMA Initiative, the "cork" was to one of the Island's many ancient visitors. Their curiosity got the best of them and they penetrated the Island's life-force, nearly damning all of humanity in the process. Using some ancient method, they were able to seal away the force with this stone cork and water system, just like DHARMA did with the hatch computer and its mechanisms.

5. Why did certain people (Walt, Miles, Hurley, etc.) have powers?
As all life stems from the Island's life-force, some are naturally more in tune with it than others. Miles was born near the source of that life-force, while Walt was likely born close to one of the hotspots connected to it. Others, like Richard and Hurley, were bestowed with abilities by a guardian, who themselves have the ability to control the nature of life as they wish.

6. What was the significance of electromagnetism?
Electromagnetism was one of the ways the life-force hidden away beneath the Island manifested itself to scientists.

7. Why did the Man in Black turn into black smoke when he was thrown into the cave?
The life-force on the Island is dualistic. Needing to protect itself from humanity, it makes a distinction between light and dark. On the light side are those who understand the risks of knowing too much, and who are willing to protect the world by protecting the Island and not allowing it to be exploited. On the dark side are those who wish to learn the Island's secrets and exploit them to their own ends. When one is "marked" with the dark side they are doomed to do the opposite of what they wish: as opposed to leaving the Island and putting it in danger, they are doomed to stay there and protect it. This marking is a very specific rite of judgment that essentially robs the marked of their humanity and their free will, as well as their ability to fight back. So when Jacob threw the Man in Black into the cave it wasn't the properties of it that changed him, it was the act of Jacob meting out this judgment and forcing him into thralldom as the preternatural laws of the Island dictate as being the punishment for what he had done.

8. Why couldn't the Man in Black hurt Jacob or the candidates?
This is one of the laws dictated by the life-force. When one is made guardian they are able to manipulate the laws of the universe, so one saying "this person can't hurt me" is no more breakable than the laws of thermodynamics.

9. What is the meaning of the numbers?
The interconnectedness of everything. At the deepest mythological level, they represent the life-force of the universe letting its will be known via the rivers of energy that flow throughout the world. They represent the numbers assigned to the candidates on the cave wall, or the Valenzetti Equation and what's at stake if the Island isn't protected. They are literally destiny popping up in everyone's lives and saying, "Hey, see these patterns? Pay close attention to them, because this is where your life is headed, whether you like it or not." And if that sounds fickle or vincenty, that's kind of the point.
10. Why did the Others want peoples' blood?
They were hoping to find a biological cause or solution to the fertility issue.

11. Where did Hurley get his nickname from?
The idea was that he sometimes ate until he puked.

12. What is the Island?
One of the earliest contributions Abrams and Lindelof made to the show bible after taking the reins from Lieber:

After the big bang/creation event, the life-force/singularity propelled its vestigial components outward across the cosmos. Its heart was left intact, cooling at the center of the universe until it became encased in a shell. While the vestigial components (other planets/moons/stars,etc.) were without life, the life-force's shell flourished. Life sprang up from the "veins" of this entity (ley lines/hotspots) all throughout the world, and over time it came to be known as Earth. As humanity began to grow and learn, they also became curious. Through religion and mysticism they would seek out the mysteries of the universe. In time, they would progress and do the same with science and research.

The true heart of the life-force lay beneath an island in the pacific. Being the universe's primeval life-blood in its rawest, purest form, it knew that it must protect itself. Though humanity would inevitably tap into its veins and plunder its power in the name of magic and later energy, and in the process come to know it as god or even simply nuclear power, it was important that the source of these things never be found. So the life-force hid itself behind a cloak of power and called to itself a guardian.

The guardian was given the power to manipulate the life-force at will, rewriting the laws of nature at its very whim. Over time it grew lonely and curious and decided to draw other people to itself, hoping to pass on its knowledge and power and finally rest. As others arrived though, the guardian found that humanity's curiosity was its greatest weakness. Whether it be through religion or science, there were always those who would seek out the life-force solely to exploit it, whether it be through control or destructive power. These would be marked with darkness, doomed forever to protect the life-force against their will, unable to escape its grasp or harm its guardians. Then there were those who would be marked with light, protectors of life deemed worthy to carry on the role of guardian, until the time came to move on.

Our story starts amidst one of these struggles. Generations have come and gone, and as the name of the show implies, the Circle is about to be drawn again.